Numerical Methods 2079

Section A
Long Answer Questions
Attempt any TWO questions. [2 × 10 = 20]

1. Define true error and relative error. Derive the bisection method for solving non-linear equation and using this method solve 2x^3 - 2x - 5 = 0 with initial x0 = 1 and x1 = 2. Calculate up to 10th iteration. [2+4+4]

2. What are the applications of interpolation? Differentiate between interpolation and regression. Consider the following data points, estimate f(10) using Lagrange's interpolation.

3. What do you mean by numerical integration? Write any one application of numerical integration. Write an algorithm and C program to implement multi-segment trapezoidal rule. [1+1+4+4]

Section B
Short Answer Questions
Attempt any EIGHT questions. [8 × 5 = 40]

4. Construct the divided difference table for the following data value and find first and second order derivative at x = 2.

55. Why is it better to use composite Simpson's 3/8 rule instead of Simpson's 3/8 rule? Find the value of integration for the following data set using Simpson's 3/8 rule.

6. What is the least squares method of fitting a function? Fit the second order polynomial for the following data values.

7. Solve the following system of linear equations using Gaussian elimination method.
2x + 2y + z = 12
3x + 2y + 2z = 8
5x + 10y - 8z = 10 [5]

8. Solve the following system of linear equations using Gauss-Seidel method.
10x + y + z = 12
2x + 10y + z = 13
2x + 2y + 10z = 14 [5]

9. Define eigenvalue and eigenvector. Explain how the shooting method is used to solve boundary value problems. [2+3]

10. Find the approximate value of y when x = 0.6 of dy/dx = 1 - 2xy, given that y = 0 when x = 0 with h = 0.2 using Heun's method. [5]

11. Consider a steel plate of size 24 cm × 24 cm. If two of the opposite sides are held at 100 degrees Celsius and the other two opposite sides at 0 degrees Celsius, find the steady state temperatures of interior points, assuming a grid size of 8 cm × 8 cm. [5]

12. Write an algorithm for Horner's method. Evaluate the polynomial f(x) = x^4 + 3x^3 + 5x^2 + 7x + 9 at x = 2 by using Horner's method. [5]